By [1]} there exists a subsequence of \((u_i)\)  that converges pointwise almost everywhere.
Thus we can choose a subsequence \((v_k) := (u_{i_k})\) , such that \((v_k)\)  converges pointwise almost everywhere, and
\(\Vert  v_{k+1} - v_k \Vert _{1,\varphi }< 4^{-k}\) 
for every \(k \in \mathbb {N}\) .
For every \(k \in \mathbb {N}\) , let \(f_k:= 2^k(v_{k+1} - v_k)\in C^1(\Omega ) \cap W^{1,\varphi }(\Omega )\) .
For every \(j \in \mathbb {N}\) , let
\(g_j:= \sum _{k=1}^j |f_k|\quad \text{and}\quad h_j := \sum _{k=1}^j |\nabla f_k|.\) 
Since the sequences \((g_j(x))\)  and \((h_j(x))\)  are increasing for every \(x \in \Omega \) , the limits \(g(x) := \lim _{j\rightarrow \infty } g_j(x)\)  and \(h(x) :=\lim _{j\rightarrow \infty } h_j(x)\)  (possibly \(\infty \) ) exist.
Since the functions \(g_j\)  are continuous, \(g\)  is a Borel function.
If \(j<m\) , then by [1]}
\(\Vert g_m -g_j\Vert _{\varphi }\lesssim \sum _{k=j+1}^m \Vert f_k\Vert _{\varphi }\leqslant \sum _{k=j+1}^\infty \Vert f_k\Vert _{1,\varphi }< \sum _{k=j+1}^\infty 2^{-k}= 2^{-j},\) 
which implies that \((g_j)\)  is a Cauchy sequence in \(L^{\varphi }(\Omega )\) .
By Lemma REF , \(g\)  is the limit of \((g_j)\)  in \(L^\varphi (\Omega )\) .
Similarly, since
\(\Vert h_m-h_j\Vert _\varphi \lesssim \sum _{k=j+1}^m \Vert \nabla f_k\Vert _{\varphi }\leqslant \sum _{k=j+1}^\infty \Vert f_k\Vert _{1,\varphi }< 2^{-j},\) 
we find that \(h\)  is the limit of \(h_j\)  in \(L^\varphi (\Omega )\) .
Since \(f_k \in C^1(\Omega )\) , for any \(k\in \mathbb {N}\)  we have
\(\big {|}|f_k(x)| - |f_k(y)|\big {|}\leqslant |f_k(x) - f_k(y)|\leqslant \int _\gamma |\nabla f_k| \,ds\) 
for every \(x,y\in \Omega \)  and any \(\gamma \in \Gamma _{rect}(\Omega )\)  containing \(x\)  and \(y\) .
Thus for every \(j \in \mathbb {N}\)  we have
\(|g_j(x)-g_j(y)|\leqslant \sum _{k=1}^j \big {|}|f_k(x)| - |f_k(y)|\big {|}\leqslant \sum _{k=1}^j \int _\gamma |\nabla f_k| \,ds= \int _\gamma h_j \,ds,\) 
for every \(x,y\in \Omega \)  and any \(\gamma \in \Gamma _{rect}(\Omega )\)  containing \(x\)  and \(y\) .
Denote by \(E\)  the set of points \(x \in \Omega \)  such that the sequence \((v_k(x))\)  does not converge.
Since \((v_k)\)  converges pointwise almost everywhere, we have \(|E| = 0\) .
It is easy to see that if \(x \in E\) , then \(x \in \lbrace |f_k| > 1\rbrace \)  for infinitely many \(k \in \mathbb {N}\) , and therefore \(g(x) = \infty \) .
Thus
\(E\subset E_\infty := \lbrace x \in \Omega : g(x) = \infty \rbrace ,\) 
and \(\Gamma _E \subset \Gamma _{E_\infty }\) .
Next we construct a set \(\Gamma \subset \Gamma _{rect}(\Omega )\)  such that
\(\Gamma _{E_\infty } \subset \Gamma \)  and \(\operatorname{M_{\varphi }}(\Gamma ) = 0\) .
It then follows by Lemma REF  (a) that \(\operatorname{M_{\varphi }}(\Gamma _E) = \operatorname{M_{\varphi }}(\Gamma _{E_\infty }) = 0\) .
By Lemma REF , considering a subsequence if necessary, we find an exceptional set \(\Gamma _1 \subset \Gamma _{rect}(\Omega )\)  such that
\(\lim _{j\rightarrow \infty } \int _\gamma h - h_j \,ds= 0\) 
for every \(\gamma \in \Gamma _{rect}(\Omega ) \setminus \Gamma _1\) .
Let
\(\Gamma _2:= \Big {\lbrace } \gamma \in \Gamma _{rect}(\Omega ) : \int _\gamma g \,ds = \infty \Big {\rbrace }\quad \text{and} \quad \Gamma _3:= \Big {\lbrace } \gamma \in \Gamma _{rect}(\Omega ) : \int _\gamma h \,ds = \infty \Big {\rbrace }.\) 
For every \(m \in \mathbb {N}\) , the function \(g/m\)  is \(\Gamma _2\)  admissible, hence \(\operatorname{M_{\varphi }}(\Gamma _2) \leqslant \Vert g \Vert _\varphi / m\) .
Thus it follows that \(\operatorname{M_{\varphi }}(\Gamma _2) = 0\) .
Similarly, we see that \(\operatorname{M_{\varphi }}(\Gamma _3) = 0\) .
Let \(\Gamma := \Gamma _1 \cup \Gamma _2 \cup \Gamma _3\) .
By Lemma REF  (b) \(\operatorname{M_{\varphi }}(\Gamma ) = 0\) .
It remains to show that \(\Gamma _{E_\infty } \subset \Gamma \) .
Suppose that \(\gamma \in \Gamma _{rect}(\Omega )\setminus \Gamma \) .
Since \(\gamma \notin \Gamma _2\) , there must exist some \(y \in \operatorname{Im}(\gamma )\)  with \(g(y) < \infty \) .
By (REF ), for any \(x \in \operatorname{Im}(\gamma )\)  and any \(j \in \mathbb {N}\)  we have
\(g_j(x)\leqslant g_j(y) + |g_j(x) - g_j(y)|\leqslant g_j(y) + \int _\gamma h_j \,ds.\) 
Since \(\gamma \notin \Gamma _1\) , it follows that
\(\lim _{j\rightarrow \infty } \int _\gamma h_j \,ds= \int _\gamma h \,ds,\) 
where the right-hand side is finite because \(\gamma \notin \Gamma _3\) . Thus we have
\(g(x)= \lim _{j\rightarrow \infty } g_j(x)\leqslant \lim _{j\rightarrow \infty } \Big {(}g_j(y) + \int _\gamma h_j \,ds \Big {)}= g(y) + \int _\gamma h \,ds< \infty .\) 
Since \(x \in \operatorname{Im}(\gamma )\)  was arbitrary, it follows that \(\gamma \notin \Gamma _{E_\infty }\) .
And since \(\gamma \notin \Gamma \)  was arbitrary, it follows that \(\Gamma _{E_\infty } \subset \Gamma \) .
